If and terms of a G.P. are m and n respectively, then write its pth term.

Question

If  and  terms of a G.P. are m and n respectively, then write its pth term.

Philoid · Accepted Answer

⇒ Let the first term be a and the common ratio be r.

∴ According to the question,

a_p+q = m.

a_p-q = n.

a_n = ar^n-1

a_p+q = a.r^p+q-1

a_p-q = a.r^p-q-1

∴ a.r^p+q-1 = m.

a.r^p-q-1 = n.

Multiplying above two equations we get

a²r^{(p+q-1+(p-q-1)} = a²r^(2p-2)

a²r^(2p-2) = m.n

(ar)^2(p-1) = m.n

∴ ar^p-1 =√m.n

⇒ P^th term is given by a.r^p-1

∴ ar^p-1 =√m.n